On the theory of collective motion in nuclei. II. Quantum theory

The collective Hamiltonian is assumed to be invariant under the orthogonal group O(A ? 1, $\text{R}$). It is shown that the quantum collective dynamics can be formulated on a coset space of the symplectic group $\text{sp}$(6, $\text{R}$) of dimension 12, 16 or 18. The first case corresponds to the collective dynamics based on closed shells and leads to a Hilbert space of analytic functions in six complex collective quasiparticle variables. Dequantization of this scheme yields the classical dynamics described in I. In the limit A ? 1 one obtains a system of s- and d-bosons with symmetry group $\text{u}$ (6) in the collective state space.     

Irreducible U(3) tensor interactions in the symplectic collective model

The potential V(β, γ) of the Bohr-Mottelson and symplectic collective models is expressed as a linear combination of U(3) irreducible tensor operators in the symplectic enveloping algebra. This many-body collective potential is then projected onto the symplectic two-body tensor operators. The projected two-body potential is shown to give results similar to the many-body potential in ²⁰Ne. Hence, in the symplectic shell model, one has obtained a collective model with two-body forces.     

Exact canonically conjugate momentum to the quadrupole tensor and a microscopic derivation of the nuclear collective hamiltonian

Two momenta conjugate to the mass quadrupole tensor are given. The first is a canonical momentum only in a subspace of the shell model space. A microscopic collective kinetic energy in terms of this momentum and the quadrupole tensor is then obtained and compared with that of Bohr's hamiltonian. The second momentum is, on the other hand, canonically conjugate to the quadrupole tensor in the entire state space.     

The moment map and collective motion

Given a Hamiltonian action of a Lie group G on a symplectic manifold M there is an induced map Φ: $\text{M → g}{}^1$ where $\text{g}{}^1$ is the dual space to the Lie algebra, g, of G. The map Φ is called the moment map. Any function P on $\text{g}{}^1$ then gives rise to a function F = P ° Φ on M which is a “collective Hamiltonian” associated to the group action G. We show how the rigid rotor, liquid drop, and other collective models of the nucleus fit into this framework. We describe the steps involved in integrating collective equations of motion and indicate some principles involved in the choice of collective Hamiltonians, i.e., the functions P. We discuss these constructions in some detail for the case that G is a semidirect product.     

Analytic expressions for energy centroids and widths in the microscopic collective model

Analytic formulae are given for the U(3) centroids of the collective Bohr-Mottelson potential in the microscopic collective model. In particular, formulae are reported for the centroids of the quadratic [Q · Q ～ β²] and cubic [Q · (Q × Q ～ β³cos 3γ] rotational scalars in the microscopic quadrupole operator. Favorable comparisons for ground-state intensities are achieved between shell-model diagonalizations and statistical predictions based upon the gaussian approximation to the energy density. These results suggest that statistical measures can be used reliably for truncation of the infinite-dimensional representation spaces of the microscopic symplectic collective theory.     

The generator-coordinate-method with conjugate parameters and the unification of microscopic theories for large amplitude collective motion

A dynamic theory of large amplitude collective motion of many particle systems is presented which is relevant, for example, to nuclear fission. The theory is microscopic and makes use of a collective path, i.e. a suitably constructed set of distorted nonequilibrium Slater determinants. The approach is a generalization of the generator coordinate method (GCM) and improves its dynamic aspects by extending it to a pair of conjugate generator parameters q and p (DGCM). The problems connected with redundancy and superfluous degrees of freedom are solved by prediagonalizing the local oscillations about each point of the dynamic collective basis | q, p ～. For adiabatic large amplitude collective motion a Schrödinger equation is derived which appears to be nearly identical to the one obtained by a consistent quantization of semiclassical approaches as e.g. the adiabatic time dependent Hartree-Fock theory (ATDHF). In turn a collective path constructed by ATDHF proves to be particularly suited for being used in the present DGCM formalism. Altogether the formalism unifies two classes of microscopic approaches to collective motion, viz. the quantum mechanical GCM and the classical theories like cranking and ATDHF.     

Excitation during collective deformation: How simple it is

In the frame of the time dependent Hartree-Fock-Bogolyubov method a simple model for the excitation of a nucleus during a collective deformation is presented. The Hilbert space consists of all single and multiple pair excitations the nucleons of each pair being in time reversed states. The non-adiabatic transitions within this space are treated correctly. Treating the selfconsistency ofλ andΔ only in an average way reduces the multichannel calculation to a set of Landau-Zener problems. These give the excitation probability of each pair. Introducing average quantities yields an analytic solution for the excitation energy in the fission process, the collective kinetic energy at the scission point and the mean number of pair excitations for any deformation.     

On the theory of collective motion in nuclei. I. Classical theory

From the assumption that the collective Hamiltonian be invariant under the orthogonal group O(A ? 1, $\text{R}$) it is concluded that classical collective dynamics can be formulated on a symplectic manifold. This manifold is shown to be a coset space of the symplectic group $\text{L}$h(6, $\text{R}$) of dimension 12, 16 or 18. The first case corresponds to the dequantization of closed-shell collective dynamics and is described in terms of six complex s- and d-quasiparticles. In the limit A ? 1 it is shown that a transformation leads to interacting s- and d-bosons with the symmetry group $\text{u}$ (6) in the collective phase space.     

Entanglement and localization of a two-mode Bose-Einstein condensate

A simple second quantization model is used to describe a two-mode Bose-Einstein condensate (BEC), which can be written in terms of the generators of a SU(2) algebra with three parameters. We study the behavior of the entanglement entropy and localization of the system in the parameter space of the model. The phase transitions in the parameter space are determined by means of the coherent state formalism and the catastrophe theory, which besides let us get the best variational state that reproduces the ground state energy. This semiclassical method let us organize the energy spectrum in regions where there are crossings and anticrossings. The ground state of the two-mode BEC, depending on the values of the interaction strengths, is dominated by a single Dicke state, a spin collective coherent state, or a superposition of two spin collective coherent states. The entanglement entropy is determined for two recently proposed partitions of the two-mode BEC that are called separation by boxes and separation by modes of the atoms. The entanglement entropy in the boxes partition is strongly correlated to the properties of localization in phase space of the model, which is given by the evaluation of the second moment of the Husimi function. To compare the fitness of the trial wavefunction its overlap with the exact quantum solution is evaluated. The entanglement entropy for both partitions, the overlap and localization properties of the system get singular values along the separatrix of the two-mode BEC, which indicates the phase transitions which remain in the thermodynamical limit, in the parameter space.     

Microscopic study of fermionic collective motion: (I). Functional representation of collective motion

Based on the shell structure of the finite nuclear many fermion system (FMFS), the coherent states related to the Spin(2r) group are defined. The global and local functional representations of the FMFS state-vectors and operators, defined on the coset space Spin(2r)/U(r), are constructed. The nonuniqueness of the coherent state functional representations is overcome by the imposition of a consistency condition on the wave functions. The influence of the boundary of the coset space Spin(2r)/U(r) on the local functional representation is physically removed only for the bound states of FMFS. The reason for the non-hermitian behavior of the local functional representation is exposed. Finally, using Bargmann's theory, the boson representation of FMFS are directly calculated from the local functional representation of FMFS. Thus, in this paper, we have demonstrated that the kinematics of the collective behavior of FMFS can be described in three non-equivalent representations: the fermion representation, the global functional representation and the local functional representation.     

Microscopic description of the collective states in N = 70 isotones on the IBM basis

Low-lying collective quadrupole states in four N = 70 isotones are analyzed within interacting boson model (IBM1) using parameters calculated on the basis of a microscopic quasiparticle theory that considers variation of the superconducting and collective quadrupole properties with increasing excitation energies and spins.     

Q-phonon scheme in the collective nuclear model

The Q-phonon scheme developed on the basis of the algebraic collective nuclear model is discussed. It is shown that, within this scheme, low-lying collective states of even-even nuclei can be described to a precision higher than 90% of the norm by using one or, at maximum, two components of the Q-phonon basis constructed by applying a fixed number of the quadrupole operators Q to the exact ground state of the system. Various applications of this approximate scheme are discussed. It is shown that, by using this scheme, the relations between several E2-transition probabilities or between the energies of the collective states can be derived. It is also shown that the Q-phonon scheme can be used to extract information about the equilibrium shapes of nuclei and their fluctuations from data on the E2-transition probabilities.     

Dual squeezed states in an atom-photon cluster and their manifestations

The general kinetic equation for an isolated two-level atom and a high-Q cavity mode in a heat bath exhibiting quantum correlations (entangled bath) is applied to the analysis of the squeezed states of the collective system. Two types of collective operators are introduced for the analysis: one is based on bosonic commutation relations, and the other, on the commutation relations of the algebra obtained by a polynomial deformation of the angular momentum algebra. On the basis of these relations, formulas for observables are constructed that identify squeezed states in the system. It is shown that, under certain conditions, the collective system exhibits dual squeezing within the relations for boson operators, as well as for the operators constructed from the angular momentum algebra. Such squeezing is demonstrated under a projective measurement of an atom and for an entanglement swapping protocol. In the latter case, when measuring two initially independent atomic systems, depending on the type of measurement, two cavity modes collapse into a nonseparable state, which is described either by a nonseparability relation based on boson operators or by a relation based on the operators of the algebra of the quasimomentum of the collective system consisting of these two modes.     

Level density and collective enhancement factor of a compound nucleus in non-adiabatic approach

In this paper a simple statistical formalism that takes into account the diabatic motion of a compound system to calculate total level density is given. The diabatic motion is introduced by coupling the adiabatic collective motion to an environment consisting of intrinsic degrees of freedom. The collective enhancement of level density is studied. It has anadiabatical coefficient that depends only on the excitation energy and adynamical one that depends on the excitation energy and the coupling between intrinsic and collective modes. Qualitatively, the coefficients of the vibrational enhancement factor forA=240 have been studied. The coupling is understood as the initial effect of the oscillatory motion of the mass asymmetry of fragments on the nucleonic motion. The damping effect of the vibrational motion is taken into account considering the collective motion of the fragments as a zero-sound wave propagating in the Fermiliquid. The intrinsic state in the framework of the Fermi Gas Model (FGM) is described. The Pashkevich’s parametrization is used to describe the binary decay of the compound nucleus.